Polytope of Type {66,4} with group SmallGroup(1584,657)

Overview

Group: SmallGroup(1584,657)
Rank: 3
Schläfli Type: {66,4}
Vertices, edges, …: 198, 396, 12
Order of s₀s₁s₂: 44
Order of s₀s₁s₂s₁: 6
Also known as: if this polytope has a name.

Special Properties

Compact Hyperbolic Quotient
Locally Spherical
Orientable

Quotients maximal quotients in bold

9-fold

{22,4}*176

11-fold

{6,4}*144

18-fold

{22,2}*88

22-fold

{6,4}*72

36-fold

{11,2}*44

99-fold

{2,4}*16

198-fold

{2,2}*8

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

P/N, where N=<(s0*s1)^7*s0*s2*(s1*s0)^3*s2*s1> of order 2

7 facets

5 of {66}*132
2 of {33}*66

99 vertex figures

99 of {4}*8

P/N, where N=<(s0*s1)^3*s2*(s1*s0)^2*s1*s2> of order 2

6 facets

6 of {66}*132

99 vertex figures

99 of {4}*8

P/N, where N=<s0*s1*s2*(s1*s0)^2*s1*s2*s1*s0*s2*s1*s2> of order 3

8 facets

2 of {66}*132
6 of {22}*44

66 vertex figures

66 of {4}*8

P/N, where N=<(s0*s1*s2*s1)^2> of order 3

4 facets

4 of {66}*132

66 vertex figures

66 of {4}*8

P/N, where N=<(s0*s1)^2*(s2*s1*s0)^2*(s1*s2)^2, (s0*s1)^8*s0*s2*(s1*s0)^2*s1*s2> of order 6

5 facets

3 of {22}*44
2 of {33}*66

33 vertex figures

33 of {4}*8

Representations

Permutation Representation (GAP)

s0 := ( 2,11)( 3,10)( 4, 9)( 5, 8)( 6, 7)(12,23)(13,33)(14,32)(15,31)(16,30)(17,29)(18,28)(19,27)(20,26)(21,25)(22,24)(34,67)(35,77)(36,76)(37,75)(38,74)(39,73)(40,72)(41,71)(42,70)(43,69)(44,68)(45,89)(46,99)(47,98)(48,97)(49,96)(50,95)(51,94)(52,93)(53,92)(54,91)(55,90)(56,78)(57,88)(58,87)(59,86)(60,85)(61,84)(62,83)(63,82)(64,81)(65,80)(66,79);;
s1 := ( 1,35)( 2,34)( 3,44)( 4,43)( 5,42)( 6,41)( 7,40)( 8,39)( 9,38)(10,37)(11,36)(12,46)(13,45)(14,55)(15,54)(16,53)(17,52)(18,51)(19,50)(20,49)(21,48)(22,47)(23,57)(24,56)(25,66)(26,65)(27,64)(28,63)(29,62)(30,61)(31,60)(32,59)(33,58)(67,68)(69,77)(70,76)(71,75)(72,74)(78,79)(80,88)(81,87)(82,86)(83,85)(89,90)(91,99)(92,98)(93,97)(94,96);;
s2 := (12,34)(13,35)(14,36)(15,37)(16,38)(17,39)(18,40)(19,41)(20,42)(21,43)(22,44)(23,67)(24,68)(25,69)(26,70)(27,71)(28,72)(29,73)(30,74)(31,75)(32,76)(33,77)(56,78)(57,79)(58,80)(59,81)(60,82)(61,83)(62,84)(63,85)(64,86)(65,87)(66,88);;
poly := Group([s0,s1,s2]);;

Finitely Presented Group Representation (GAP)

F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1, 
s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s2 ];;
poly := F / rels;;

Permutation Representation (Magma)

s0 := Sym(99)!( 2,11)( 3,10)( 4, 9)( 5, 8)( 6, 7)(12,23)(13,33)(14,32)(15,31)(16,30)(17,29)(18,28)(19,27)(20,26)(21,25)(22,24)(34,67)(35,77)(36,76)(37,75)(38,74)(39,73)(40,72)(41,71)(42,70)(43,69)(44,68)(45,89)(46,99)(47,98)(48,97)(49,96)(50,95)(51,94)(52,93)(53,92)(54,91)(55,90)(56,78)(57,88)(58,87)(59,86)(60,85)(61,84)(62,83)(63,82)(64,81)(65,80)(66,79);
s1 := Sym(99)!( 1,35)( 2,34)( 3,44)( 4,43)( 5,42)( 6,41)( 7,40)( 8,39)( 9,38)(10,37)(11,36)(12,46)(13,45)(14,55)(15,54)(16,53)(17,52)(18,51)(19,50)(20,49)(21,48)(22,47)(23,57)(24,56)(25,66)(26,65)(27,64)(28,63)(29,62)(30,61)(31,60)(32,59)(33,58)(67,68)(69,77)(70,76)(71,75)(72,74)(78,79)(80,88)(81,87)(82,86)(83,85)(89,90)(91,99)(92,98)(93,97)(94,96);
s2 := Sym(99)!(12,34)(13,35)(14,36)(15,37)(16,38)(17,39)(18,40)(19,41)(20,42)(21,43)(22,44)(23,67)(24,68)(25,69)(26,70)(27,71)(28,72)(29,73)(30,74)(31,75)(32,76)(33,77)(56,78)(57,79)(58,80)(59,81)(60,82)(61,83)(62,84)(63,85)(64,86)(65,87)(66,88);
poly := sub<Sym(99)|s0,s1,s2>;

Finitely Presented Group Representation (Magma)

poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1, 
s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s2 >;

References

None.

to this polytope.